Improper fitment of the action in the stock will cause this.
Torquing down the front action screw- in your case- appears to be causing the barrel to contact, or change location of contact, of the barrel in the forend.

In any case, front and rear screws should always be fully tightened, and 1/4 turn should never happen- it's just plain loose if you can tighten that much. The amount of torque for best accuracy is always a variable, but that's "fine tuning".

You say poi shifts (as would be expected)- but does it remain consistent with the front screw properly torqued? 30-40 in/lbs. should be about right for a wood stock without pillars.

Russian/Soviet 91/30 stocks often have terrible inletting (compared to the Finns).
Every rifle is different, some like to create a pressure point as in the pictures, others prefer to free-float. Requires some experimentation, try different methods using a piece of paper to check for locations of barrel contact, and see what provides the tightest groupings.

"...wandering zero after removing it from the stock..." Every time you do that you alter the bedding.
"...do some basic checks..." Leave the stock on with the screws tight, sight in and leave the stock alone.
Mosins are not target rifles either.

My 91/30 has always seemed to have a wandering zero after removing it from the stock.

At the range I am able to shift the POI up 4" at 50 yards by tightening the front action screw 1/4 turn!

How can I do some basic checks on the action/barrel/stock bedding and fit?

Every Mosin Nagant I owned, which I then bedded, shot consistently and shot round groups. I was limited by barrel quality and bullet quality.

Whatever bedding these rifles were supposed to have, they no longer have after 70-100 years of use, and a couple of rebuilds. Every Mosin, including the Finnish Mosin Nagants, the wood had compressed, if it was in the original stock, and if it was not in the original stock, the action was severely bowed.

I just routed out some wood around the front of the action and the rear tang. Coated all metal parts with Johnson paste wax, as a release agent, and poured in a good two part epoxy. I used Acura Glass gel on my Nagants, today I would use whatever I have on the shelf, which is Tex Marine. These rifles are not accurate enough or worth enough to spend the big bucks bedding them with Devcon Aluminum or Stainless steel epoxy.

Got to tell you, with the short 20" barrel on this M38

with each shot, it produced One Big Badda Boom! You can see fireballs in direct sunlight.

It did seem a bit loose in the stock. You could grab the barrel and see it and the stock wiggle around each other.

Instead of oiled felt I glued a thin piece of cardboard to the stock near the front.

On the back I glued one brass shim, but noticed front action screw ended in the same place as before for the same torque setting. Then I tripled the shim thickness and now it reaches 25 in lbs about half a turn earlier.

Everything fits tight now, so I'm going to hit the range and see what happens.

Also Google “corking” a Mosin Nagant. Put a small strip of cork under the hand guard as close to the muzzle as you can, so the barrel pressure is more consistent. Some people swear by it, some people try to full free-float the thing.

__________________Those who hammer their swords into plow shares will plow for those who didn't...

The first shot was much lower than the rest, which makes since because I think the brass shims I glued in needed to compress a bit. The next four were pretty tight. I think I got this solved. Previously the action screws would work loose easily but they stayed tight.

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